Evaluate $~~\int xe^{-x}dx\,$. Choose 1 answer: Choose 1 answer: (Choice A) A $ -e^{-x}(x-1)+C$ (Choice B) B $e^{-x}(x+1)+C$ (Choice C) C $ e^{-x}(x-1)+C$ (Choice D) D $-e^{-x}(x+1)+C$
Explanation: We will solve this by integrating by parts. We know that $ \int u(x)v\,^\prime(x)dx = u(x)v(x)-\int u\,^\prime(x)v(x)dx\,$. We can rewrite this as $ \int u\ dv = uv-\int v\ du\,$. In this problem we will let $~u = x~$ and $~dv=e^{-x} dx\,$. Then $~du = dx~$ and $~v = \int e^{-x}dx = -e^{-x}\,$. Integration by parts gives $ \,\int xe^{-x}dx = x\cdot\big(-e^{-x}\big)-\int-e^{-x}\,dx$ $ ~= (-xe^{-x})-e^{-x}+C $ $~ = -e^{-x}(x+1)+C$